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If there are $16$$8$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the open interval $\left(0,1\right)$, what is the expected largest size of a subset in which the points form the vertices of a convex polygon? Thanks!

If there are $16$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the open interval $\left(0,1\right)$, what is the expected largest size of a subset in which the points form the vertices of a convex polygon? Thanks!

If there are $8$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the open interval $\left(0,1\right)$, what is the expected largest size of a subset in which the points form the vertices of a convex polygon? Thanks!

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What is the expected value for this

If there are $16$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the open interval $\left(0,1\right)$, what is the expected largest size of a subset in which the points form the vertices of a convex polygon? Thanks!